Introduction
The 3D Pipe Routing (PR) problem is a common industrial problem that appears when designing the layout of industrial plants, such as oil refineries, natural gas processing stations, water treatment facilities, and the type of power plants used in ships and submarines. Designing the layout of such a plant requires finding 3D location coordinates for every piece of equipment in the plant (equipment allocation problem), and finding a 3D route for every pipe that connects two pieces of equipment (PR problem). The aim is to minimize the total cost of the plant (which can run into multibillion dollar budgets), while ensuring safety and correct functionality. Figure 1 shows a layout for part of a natural gas plant.
Differences in the quality of the final layout, can have a very significant impact on the cost of these plants, including the cost of the pipes and associated support structures, which are known to take the largest share: up to 80% of the purchased equipment cost or 20% of the fixedcapital investment (Peters and Timmerhaus, 2004). However, finding highquality plant layouts is remarkably difficult due to the size of these plants and the complexity of the associated constraints. As a result, layouts are still designed manually, taking multiple engineers many months (or even years) to complete. This process is inefficient, costly and the results may vary in quality, since they largely depend on the experience of the piping and layout engineers.
Current research into automatic plant layout commonly divides it into two phases. The first phase performs equipment allocation, that is, finds 3D positions for all equipment that minimise a total cost and satisfy all equipment constraints, such as min/max distances and maintenance access requirements. In this phase the cost of the pipes is approximated using rough measures, such as Manhattan distances. The second phase solves the PR problem, that is, finds 3D routes for all pipes connecting the (already allocated) equipment, that minimize the pipe costs (based on their length) and satisfy all pipe constraints, such as no two pipe routes collide and they are all appropriately supported. In such setting, the start and end position of each pipe is given as input (referred to as nozzles), representing the pipe’s connection to its source/target equipment.
The PR problem is similar to the 2D MultiAgent Path Finding (MAPF) problem, which searches for collisionfree paths for several agents from given start locations to given goal locations in a known 2D environment. Thus, a solution to a MAPF instance is a set of blocked cells in xyt space, while a solution to the corresponding PR instance is the corresponding set of blocked cells in xyz space. In this paper, we show how to use this similarity to apply several recently developed optimal and boundedsuboptimal MAPF algorithms to the PR problem. We also discuss the performance of some of these algorithms on different sets of PR instances. Finally, we discuss further research necessary to tackle realworld piperouting instances of interest to industry today. This opens up a new direction of industrial relevance for the MAPF research community.
The Core Pipe Routing Problem
This section defines an abstract version of the PR problem, which will be used throughout the rest of the paper. The abstraction uses a discretization of the plant’s volume into a unitcost 6neighbor 3D grid graph. In this graph, each vertex is associated with a coordinate , and represents either an open or a blocked grid cell at that coordinate. A unitcost edge between two vertices exists only if both vertices are open. Along with this graph, we are given a set of pipes. Each pipe specifies its start and goal vertices, and , respectively. A solution is a set of routes connecting each pipe’s start and goal vertices. A route is simply a path (that is, a sequence of edges) in the abstract graph. A solution is feasible if each vertex (and thus each edge) is used to route at most one pipe. The cost of a solution is the sum of route lengths. Finding a solution with minimum cost is NPhard (due to the equivalence with 2D MAPF for the objective function of the sum of individual costs, which is known to be NPhard, Yu and LaValle 2013).
While the above formulation abstracts away some aspects of the realworld problem (such as pipe stress and support requirements) it maintains the core combinatorial hardness. This is because in a feasible solution pipes may need to accommodate each other and, thus, routed differently than their individual shortest paths.
MAPF Algorithms for Core Pipe Routing
In this paper we solve Core Pipe Routing using two different MAPF algorithms. The first one is ConflictBased Search (CBS) (Sharon et al., 2015), an optimal MAPF solver that performs highlevel and lowlevel searches. Each highlevel node contains a set of constraints and, for each agent (we will use the terms agent and pipe interchangeably), a feasible path that respects these constraints. A highlevel node is a goal node if and only if none of its paths collide. The highlevel root node has no constraints. The highlevel search of CBS is a bestfirst search that defines the value of a highlevel node as their cost, that is, as the sum of the travel times along the pipe paths it contains. Figure 2 shows an example. When CBS expands a highlevel node , it checks whether is a goal node. If it is, CBS terminates successfully and outputs the paths in as solution. Otherwise, at least two paths collide. CBS chooses a collision to resolve and generates two highlevel child nodes of . Both child nodes inherit the constraints of and each has an additional constraint that resolves the chosen collision. During the generation of the highlevel node , CBS performs a lowlevel search for the agent affected by the newly added constraint. The lowlevel search for agent is a (bestfirst) A* search that ignores all other agents and finds a minimumcost path from the start vertex of agent to its goal vertex that respects the constraints of that involve agent .
The second algorithm is ECBS() (Barer et al., 2014), a suboptimal variant of CBS whose highlevel and lowlevel searches are focal searches (Pearl and Kim, 1982), rather than bestfirst searches. Like A*, focal searches use an OPEN list of nodes sorted in increasing order of their values. Unlike A*, focal searches with suboptimality factor also use a FOCAL list of all nodes currently in OPEN, whose values are no larger than times , the smallest value in the current OPEN. The nodes in FOCAL are sorted in increasing order according to secondaryheuristic values. While A* expands a node in OPEN with the smallest value, a focal search expands a node in FOCAL with the smallest secondary heuristic value. Secondary heuristic values do not have to be consistent (or even admissible). The highlevel and lowlevel focal searches of ECBS() use measures related to the number of collisions, as secondary heuristic values. Algorithm 1 shows a version of ECBS() adapted to the Core Pipe Routing problem by removing the temporal aspect.
Experimental Setup
For our feasibility study we generate a wide range of Core Pipe Routing Problem instances, and solve them using variants of some well known MAPF algorithms. Our experiments attempt to capture one or more of the main challenges associated with routing pipes in practice: many pipes, many components that must be avoided, small congested spaces and a requirement that paths be computed with a high degree of precision.
Environments:
Our test environments comprise threedimensional cubes made up of square cells. We generate two different types of cubes and two different sizes for each, all as follows:

Empty environments: these are a small cube of size and a large cube of size . Each cell in each empty cube is considered traversable. The smaller cube is selected to evaluate performance and scalability (in terms of number of pipes) in congested spaces. The larger cube, meanwhile, is selected to evaluate settings of realistic size, which often contain many millions of cells to explore.

Obstacle environments: these are again two cubes of the same size as the empty environments, where we introduce random obstacles. These environments are selected to evaluate performance and scalability in settings where there exist many pieces of equipment that must be avoided. Obstacles take the form of columns, which originate from the floor of the environment (i.e. from a cell at height = 0), and rise up to a randomly selected height. The origins of these columns are also selected at random to comprise 10% of the cells in each cube.
Instances:
For each environment we select at random pairs of traversable cells (resp. start and target locations) from among those appearing on the perimeter of each cube. In other words, we assume pipes begin and end from random positions on the face of each cube. A set of such starttarget pairs is called an instance of the Core Pipe Routing Problem. The number of pipes per instance varies depending on the environment. For the small environments we proceed in increments of 10, up to 200 pipes. For large environments, we proceed in increments of 50, up to 650 pipes.
Note that, as with MAPF, two instances having the same number of pipes (equiv. agents) may require very different amounts of computation to solve. Depending on the positions of the start and target cells one instance may be relatively straightforward while another can require significant coordination effort. To mitigate such bias in our results, we choose to generate 50 instances per increment.
Algorithms:
We test three suitably modified MAPF algorithms on the set of generated problem instances. These are:

Conflictbased Search (CBS), which returns an optimal solution if one exists.

ECBS(1.01), which returns a solution guaranteed to be no more than 1% larger than optimal.

ECBS(1.05), which returns a solution guaranteed to be no more than 5% larger than optimal.
All algorithms are implemented in C++ and have a 100s timeout (i.e. they return failure if a solution cannot be found after this amount of time). We repeat each experiment 50 times for different number of pipes, using different randomly generated start and goal locations. Our test machine is a 3.5GHz AMD Ryzen 3 2200G desktop computer with 16GB dual channel 2400MHz RAM.
Results
We evaluate each of the three algorithms — CBS, ECBS(1.01) and ECBS(1.05) — on each of our four test environments. We analyse results as part of two distinct experiments: small environments and large environments. In each case we focus on three distinct metrics:

Success Rate, which measures the percentage of problems that were solved for each fixed number of pipes (recall that we generate 50 instances per increment). We will say that an algorithm usually succeeds if its success rate for pipes is .

Runtime, which measures, in seconds, the time required to solve an instance. Rather than computing average performance (e.g. per pipes) we focus instead on performance across the full set of instances on a given environment. The distribution of results is always presented in sorted order, from lowest runtime to highest runtime (recall that we use a 100s timeout).

Solution Quality, which measures the quality of solutions computed by ECBS(1.01) and ECBS(1.05) with respect to the best known bound (i.e. either the solution cost found by optimal CBS or the minimum value of any node in the FOCAL list).
Experiment 1: Small Environments
In this experiment we focus on optimal and boundedsuboptimal pipe routing in a small congested environment of size , both with and without obstacles. Figure 3 gives a summary. We see that: (i) CBS usually succeeds for problems of between 50  80 pipes but it times out on approximately 50% of problem instances. Meanwhile ECBS usually succeeds for problems of between 100  130 pipes and it manages to solve more than 75% of all problem instances. Because the environment is small, there is little difference between the two bounded suboptimal solvers: there exist few solutions in the FOCAL list of either method which are more than 1% from optimal. This is also reflected in the results for solution quality, where we observe few plans which are more than 1% above the best known bound.
Experiment 2: Large Environments
In this experiment we focus on pipe routing in a large environment of size 320 320 320. Such environments (with millions of cells) arise in settings that require pipes to be placed with a highdegree of precision (e.g. centimetres or less). We again consider two variants, with and without obstacles. Figure 4 gives a summary.
Here we observe that even optimal CBS can usually solve instances with more than 300 pipes. Interestingly, there is now a significant difference in runtime performance between ECBS(1.01) and ECBS(1.05), with the latter usually solving problems that approach 450 pipes. Both suboptimal variants find solutions that are within 1% of the best known lowerbound, with a small number of notable exceptions for ECBS(1.05) at the 150 and 200 pipe increments.
We observe also that runtime performance experiences a sharp phase transition. For example, between the 260th and 360th quantile of the obstacle environment, ECBS(1.05) runtime rises from a less than a second per instance to
(i.e. timeout). In other words, the algorithm quickly finds feasible solutions, when such exist, within the initial fvalue suboptimality bound, but fails when more complex coordination is required. A higher initial bound may help ECBS to find feasible solutions quickly for the timeout cases. However, more work is needed to establish whether the cost of such plans will remain significantly smaller than the promised bound.The RealWorld PR Problem
As mentioned before, this paper focuses on the second phase of the process plant layout problem, where the equipment has already been positioned safely and correctly within the plant’s volume, and the aim now is to determine the best routing for the pipes that connect the equipment.
Focusing on the principles of MAPF algorithms, in this work we have ignored most of the realworld constraints and objectives, only requiring the pipes to traverse the given feasible areas while minimizing their total length. For completeness, and to motivate further research, below we describe the most important features of the realworld PR problem and the simplifications made by the PlantLayout prototype software of (Belov et al., 2017).
The Routing Landscape. Obstacles can have any geometric shape. PlantLayout approximates them by cuboids. Circular Bends and NonAxisParallel Segments. Pipes can bend along some minimal radius. Moreover, although in practice most pipe segments are parallel to one of the coordinate axes, a few are not. PlantLayout assumes pointed bends and axisparallel segments.
Pipe Diameter. Pipes have a certain diameter and require some minimal distance to other objects. PlantLayout accurately models this.
Support Costs. A pipe has to be supported either by the ground or by some other equipment. The further it is from the supporting object or the ground, the higher the support costs. Moreover, in practice several pipes can be supported together when their routes are close. PlantLayout ignores the cost aspect by demanding that every pipe has to be at most 3 meters away from some component or the ground.
Stress and Flexibility Analysis. Pipes may contract and expand due to temperature changes in the environment and the materials they transport. This poses stress on the pipe, which needs to be accounted for using stress and flexibility analysis. There are several methods differing in their complexity and precision (ASME International, 2017).
Existing Approaches for RealWorld Pipe Routing
For small instances of plant layout design problem, Sakti et al. (2016) successfully apply an integrated approach for a satisfaction version of the problem that simultaneously allocates equipment and routes the pipes. In particular, they considered 10 equipment pieces and up to 15 pipes with 4 segments on average. However, they failed to find solutions for instances with as few as 8 pipes. Realistic plants are much larger: Belov et al. (2017) consider one of five modules in a real liquid natural gas plant (the acid gas removal module), which already has 17 equipment pieces and 27 pipes within a 381;201;216 grid graph along axes . For this and other larger, realistic problem instances that consider all equipment in the plant (containing often in excess of 100 equipment pieces and 200 pipes), integrated approaches do not scale, and the problem is divided into the two phases mentioned before: allocation and routing.
Most research into plant layout design focuses only on the equipment allocation phase (e.g., Xu and Papageorgiou 2007, 2009). While some research includes the pipe routing phase (e.g., Guirardello and Swaney 2005; Sakti et al. 2016) or even focuses on it (e.g., Zhu and Latombe 1991 and Jiang et al. 2015), existing approaches do not consider simultaneous optimal routing.
On the one hand, the more realistic approaches, which do take into account the simultaneous routing of several pipes (including branching pipes and support placement), are based on heuristic algorithms (rather than complete search methods), such as the antcolony evolutionary algorithms used by
Furuholmen et al. (2010); Jiang et al. (2015). One popular approach involves prioritised planning (Cao et al., 2018) wherein the pipes are ordered according to some fixed priority and routed in order. Each higher priority pipe then becomes an obstacle for all lowerpriority pipes. The main disadvantages of prioritised planning are suboptimality and incompleteness in general. To illustrate the limitations consider the example in Figure 5.On the other hand, the approaches that rely on complete search methods are either difficult to extend for some of the required constraints, or quickly become intractable. One of the most realistic of the complete approaches is that of Guirardello and Swaney (2005), which provides a detailed mixed integer programming (MIP) model for solving phase one, and a general overview of a networkflow MIP model for solving phase two. This second MIP model relies on the construction of a reduced connection graph that limits the possible routes of the pipes. This is used to route pipes one by one, since they suggest that simultaneous routing is too costly. While they do not give enough details regarding how the connection graph is constructed, an approach to construct such a connection graph is given, e.g., by de Berg et al. (1992), who present a higherdimensional rectilinear shortest path model that considers bend costs. A more hierarchical method using cuboid free space decomposition is given by Zhu and Latombe (1991) and applied to pipe routing. However, even if these methods are used, it is not clear how Guirardello and Swaney (2005) perform sequential pipe routing when pipes interfere with each other (Guirardello and Swaney 2005 talk about “some tuning by hand” which might be required for these cases).
Belov et al. (2017) describe a general pipe routing method that incorporates stress analysis. While their model can be applied to route several pipes simultaneously, it quickly becomes intractable. In contrast, as shown in our experiments, recent advances in MAPF allow us to solve problems with hundreds of pipes optimally or very nearly optimal (and always with a strong cost guarantee).
Conclusion
In this paper we have shown that the 3D Pipe Routing (PR) problem, which aims at placing collisionfree pipes from given start locations to given goal locations in a known 3D environment, is similar to the 2D MAPF problem. This is important because it indicates that many recently developed MAPF algorithms apply more broadly than currently believed in the MAPF research community. To demonstrate this, we have evaluated the success rate, solution quality and efficiency of three MAPF algorithms (CBS, ECBS(1.01) and ECBS(1.05)) in several different environments. Results show that MAPF algorithms are able to find solutions for large instances with optimal or near optimal quality. This provides strong incentives to the MAPF research community to perform the further research necessary to tackle realworld piperouting instances of interest to industry today.
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